calculus - how to strictly prove $sin x

$$\sin xIn most textbooks, to prove this inequality is based on geometry illustration (draw a circle, compare arc length and chord ), but I think that strict proof should be based on analysis reasoning without geometry illustration. Who can prove it? Thank you very much.

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ps:

1. By differentiation, monotonicity and Taylor formula, all are wrong, because $(\sin x)'=\cos x$ must use $\lim_{x \to 0}\frac{\sin x}{x}=1$, and this formula must use $\sin x< x$. This is vicious circle.




2. If we use Taylor series of $\sin x$ to define $\sin x$, strictly prove $\sin x

Answer

We can define $\sin x$ as power series. Applying the knowledge of power series, obtain the derivative of $\sin x$, and then we will easy prove the inequality. Concluding geometry of $\sin x$, please refer to this.